Generalized Linear Models I
Chelsea Parlett-Pelleriti
Linear Model
Linear Regression
\[ \mathbb{E}\left( y | x\right) = \mu = \mathbf{X}\beta \]
Linear Regression Assumptions
independently and identically distributed
linearity
normally distributed errors
homoskedasticity
Linear Regression Fitting
Least Squares
Maximum Likelihood Estimation
Least Squares
Maximum Likelihood Estimation
Maximum Likelihood Estimation
\[ p(y | x, \theta) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x- \mu}{2\sigma^2}} \]
In Linear Regression, the likelihood function is the normal distribution with parameters \(\theta = \left( \mu, \sigma\right)\)
We want to choose values of \(\theta\) maximize the likelihood of the data, \(\left(x,y\right )\)
Maximum Likelihood Estimation
For a single data point the value of the likelihood function, \(L\left( \theta | x_i \right)\) is:
\[ \mathcal{L} \left( \theta | y_i \right) = p(y_i | x_i, \theta) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y_i- \mu}{2\sigma^2}} \]
where \(\mu = g^{-1}(\mathbf{X}\beta)\). Because of our assumption that data points are independent (the first i in i.i.d), the likelihood value for all data points is simply the product of their individual likelihood values, since \(p(A,B) = p(A)*p(B) \text{ iff } A \mathrel{\unicode{x2AEB}} B\).
\[
\mathcal{L}\left(\theta | \mathbf{y} \right) = p(\mathbf{y} | \mathbf{x}, \theta) = \prod_{i=1}^n p(y_i | x_i, \theta) = \prod_{i=1}^n\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y_i- \mu_i}{2\sigma^2}}
\]
Maximum Likelihood Estimation
The higher the likelihood of our data, the more evidence that a particular \(\theta\) is a good fit for the data.
Maximum Likelihood Estimation
\[ \text{arg max}_{\theta} \left[ \prod_{i=1}^n\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y_i- \mu_i}{2\sigma^2}} \right] \]
to maximize, we:
- take the partial derivatives of \(\prod_{i=1}^n\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y_i- \mu_i}{2\sigma^2}}\) w.r.t. each element of \(\theta\)
- set each \(\frac{\partial}{\partial \theta_i} = 0\)
- solve (analytically, Expectation Maximization, Gradient Descent…) for \(\theta\)
But…
Maximum Likelihood Estimation
…taking the derivative of a function of products is hard, so we use log likelihood.
\[ \mathcal{l}\left(\theta | \mathbf{y} \right) = \log\left(\mathcal{L}\left(\theta | \mathbf{y} \right)\right) = \\ -\frac{n}{2} \log(2\pi) - \frac{n}{2} \log (\sigma^2) -\frac{1}{2 \sigma^2} \sum_{i=1}^n (y_i - \mu_i)^2 \]
Note: \(\log()\) is a monotonically increasing function, so choosing \(\theta\) that maximizes \(\mathcal{l}\left(\theta | \mathbf{y} \right)\) will also maximize \(\mathcal{L}\left(\theta | \mathbf{y} \right)\)
Maximum Likelihood Estimation
If we choose our errors to be normally distributed, our likelihood function for a single data point was:
\[ \mathcal{L} \left( \theta | y_i \right) = p(y_i | x_i, \theta) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y_i- \mu}{2\sigma^2}} \]
but what if we used a different distribution…?
Generalized
\[ \mathbb{E}\left (y | \mathbf{X}\right ) = \mu = g^{-1}\left (\mathbf{X}\beta \right ) \\ p\left (y|x\right ) \sim \pi\left (\mu,...\right ) \]
link function: \(g()\)
likelihood function: \(\pi()\)
Linear Models
\[ \mathbf{X}\beta \]
“linear in the parameters”
Logistic Regression as a GLM
\[ \mathbb{E}\left( y | \mathbf{X} \right) = g^{-1}\left(\mathbf{X}\beta \right) \\ g\left(\mathbf{X} \right) = log\left( \frac{p}{1-p} \right); g^{-1}\left(\mathbf{X} \right) = \frac{1}{1+e^{-x}} \\ p\left (y | \mathbf{X} \right) \sim bernoulli\left( \mathbf{X}\beta \right) \]
Logit link function
Bernoulli likelihood
Maximum Likelihood Estimation
\[ \mathcal{L}\left( \theta | \mathbf{y}\right) = \prod_{i = 1}^n p(x_i)^{y_i} + \left[1- p(x_i) \right]^{1-y_i} \]
Notice, the likelihood function uses a Bernoulli distribution rather than a Normal distribution!
Choosing a Link Function
what range of values can \(\mu\) take on?
linear regression: \(\mu\) can be any value
- identity link: \(g(x) = x\), \(g^{-1}(x) = x\)
logistic regression: \(p\) must be \([0-1]\)
- logit link: \(g^{-1}(x) = \frac{1}{1 + e^{-x}}\), \(g(x) = log \left( \frac{x}{1-x}\right)\)
Link Function Interpretation
Link Functions change the interpretation of coefficients in a GLM
logit: \(g^{-1}(x) = \frac{1}{1 + e^{-x}}\), \(g(x) = log \left( \frac{x}{1-x}\right)\)
an increase in 1 unit of \(\mathbf{X}_i\) is associated with a \(\beta_i\) increase in the log odds of \(y\)
an increase in 1 unit of \(\mathbf{X}_i\) is associated with a \(e^\beta\) times increase in the odds of \(y\)
Choosing a Likelihood Function
how are errors distributed? what are the actual values of \(y\)?
linear regression: values are continuous, distributed normally
logistic regression: values are binary, distributed bernoulli
Class
Robust Student t Regression
“fat tailed distributions, you make the rockin’ world go round”
❓ if you use a student t distribution as your likelihood function, how does the likelihood of extreme values change? What impact would that have on your regression?
Robust Student t Regression
link function: identity \(g(x) = x\), \(g^{-1}(x) = x\)
likelihood function: student-t with \(\nu\) degrees of freedom
\[ \mathbb{E}\left( y | \mathbf{X} \right) = g^{-1}\left(\mathbf{X}\beta \right) \\ g\left(\mathbf{X} \right) = \mathbf{X} ; g^{-1}\left(\mathbf{X} \right) = \mathbf{X} \\ p\left (y | \mathbf{X} \right) \sim t\left( \mathbf{X}\beta, \nu \right) \]
Robust Student t Regression
Robust Student t Regression
Robust Student t Regression
Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: Family: gaussian
Links: mu = identity; sigma = identity
Formula: y ~ x
Data: o (Number of observations: 50)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 0.26 0.20 -0.15 0.67 1.00 3877 2885
x 0.11 0.22 -0.32 0.54 1.00 4106 3117
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.48 0.15 1.21 1.80 1.00 3490 2546
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Robust Student t Regression
Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: Family: student
Links: mu = identity; sigma = identity; nu = identity
Formula: y ~ x
Data: o (Number of observations: 50)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 0.10 0.13 -0.16 0.36 1.00 3391 2690
x 0.57 0.16 0.26 0.87 1.00 3871 2665
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.77 0.13 0.55 1.05 1.00 3235 3033
nu 3.54 1.55 1.61 7.32 1.00 3741 2986
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Beta Regression
Assume \(y \in [0,1]\), what would be a problem with using linear regression to predict \(y\)?
Note: technically, beta regression is not a GLM. But if it walks like a 🦆 and quacks like a 🦆, surely…
Beta Regression
Beta Regression
link function: logit \(g^{-1}(x) = \frac{1}{1 + e^{-x}}\), \(g(x) = log \left( \frac{x}{1-x}\right)\)
likelihood function: beta distribution with mean \(\mu\)
\[ \mathbb{E}\left( y | \mathbf{X} \right) = g^{-1}\left(\mathbf{X}\beta \right) \\ g\left(\mathbf{X} \right) = log\left( \frac{p}{1-p} \right); g^{-1}\left(\mathbf{X} \right) = \frac{1}{1+e^{-x}} \\ p\left (y | \mathbf{X} \right) \sim beta\left( \mathbf{X}\beta, \kappa \right) \]
Poisson Regression
Poisson Regression works with count data.
\[ \text{PMF} = \frac{\gamma^ke^{-\gamma}}{k!} \]
Poisson Regression
link function: \(g(x) = log(x)\), \(g^{-1}(x) = e^x\)
likelihood function: Poisson \(Pois(\gamma)\)
Note: for the Poisson Distribution, \(\gamma = \mu = \sigma^2\)
Poisson Regression
Fitting GLMs in R
Linear Regression (Frequentist)
library(tidyverse)
library(brms)
library(tidybayes)
library(betareg)
# Simulate data for linear regression
set.seed(540)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
y <- 5 + 2 * x1 - 1.25 * x2 + rnorm(n)
df <- data.frame(x1,x2,y)
# Linear regression using lm/glm (Frequentist)
linear_lm <- lm(y ~ x1 + x2,
data = df)
# linear_glm <- glm(y ~ x1 + x2,
# family = gaussian(), data = df)
summary(linear_lm)
Call:
lm(formula = y ~ x1 + x2, data = df)
Residuals:
Min 1Q Median 3Q Max
-1.63137 -0.61485 0.00385 0.51727 2.43526
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.91352 0.08588 57.21 <2e-16 ***
x1 2.07372 0.09108 22.77 <2e-16 ***
x2 -1.07038 0.08441 -12.68 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8582 on 97 degrees of freedom
Multiple R-squared: 0.8743, Adjusted R-squared: 0.8718
F-statistic: 337.5 on 2 and 97 DF, p-value: < 2.2e-16Linear Regression (Interactions)
# Linear regression using lm/glm (Frequentist)
linear_lm <- lm(y ~ x1 * x2,
data = df)
linear_lm2 <- lm(y ~ x1 + x2 + x1:x2,
data = df)
summary(linear_lm)
Call:
lm(formula = y ~ x1 * x2, data = df)
Residuals:
Min 1Q Median 3Q Max
-1.60416 -0.61440 0.04543 0.49694 2.46525
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.91418 0.08621 57.001 <2e-16 ***
x1 2.06612 0.09253 22.329 <2e-16 ***
x2 -1.07044 0.08473 -12.634 <2e-16 ***
x1:x2 -0.05522 0.10399 -0.531 0.597
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8614 on 96 degrees of freedom
Multiple R-squared: 0.8747, Adjusted R-squared: 0.8708
F-statistic: 223.4 on 3 and 96 DF, p-value: < 2.2e-16
Call:
lm(formula = y ~ x1 + x2 + x1:x2, data = df)
Residuals:
Min 1Q Median 3Q Max
-1.60416 -0.61440 0.04543 0.49694 2.46525
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.91418 0.08621 57.001 <2e-16 ***
x1 2.06612 0.09253 22.329 <2e-16 ***
x2 -1.07044 0.08473 -12.634 <2e-16 ***
x1:x2 -0.05522 0.10399 -0.531 0.597
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8614 on 96 degrees of freedom
Multiple R-squared: 0.8747, Adjusted R-squared: 0.8708
F-statistic: 223.4 on 3 and 96 DF, p-value: < 2.2e-16Linear Regression (Bayesian)
# Linear regression using brm (Bayesian)
linear_brm <- brm(y ~ x1 + x2, family = gaussian(), data = df)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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Chain 4: Family: gaussian
Links: mu = identity; sigma = identity
Formula: y ~ x1 + x2
Data: df (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 4.91 0.08 4.75 5.08 1.00 4300 2774
x1 2.07 0.09 1.89 2.25 1.00 4656 3039
x2 -1.07 0.08 -1.24 -0.91 1.00 4574 3027
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.87 0.06 0.75 1.00 1.00 4404 3092
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Logistic Regression (Frequentist)
# Simulate some data for logistic regression
set.seed(540)
n <- 100
x1 <- rnorm(n)
prob <- 1 / (1 + exp(-1 - 2 * x1 + -0.2 * x2)) # Logit link
y <- rbinom(n, size = 1, prob = prob)
df <- data.frame(y,x1,x2)
# Logistic regression using glm (Frequentist)
logistic_glm <- glm(y ~ x1 + x2, family = binomial(), data = df)
summary(logistic_glm)
Call:
glm(formula = y ~ x1 + x2, family = binomial(), data = df)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.0086 0.2847 3.543 0.000396 ***
x1 1.8303 0.3863 4.738 2.15e-06 ***
x2 0.3605 0.2675 1.347 0.177857
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 126.836 on 99 degrees of freedom
Residual deviance: 88.209 on 97 degrees of freedom
AIC: 94.209
Number of Fisher Scoring iterations: 5Logistic Regression (Bayesian)
# Logistic regression using brm (Bayesian)
logistic_brm <- brm(y ~ x1 + x2, family = bernoulli(), data = df)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.7e-05 seconds
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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Chain 4: Family: bernoulli
Links: mu = logit
Formula: y ~ x1 + x2
Data: df (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 1.04 0.29 0.48 1.63 1.00 2743 2698
x1 1.93 0.40 1.20 2.78 1.00 2725 2307
x2 0.39 0.27 -0.14 0.93 1.00 2793 2876
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Poisson Regression
# Simulate some data for Poisson regression
set.seed(540)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
lambda <- exp(1 + 0.5 * x1 -0.25 * x2) # Poisson rate (log link)
y <- rpois(n, lambda)
df <- data.frame(x1,x2,y)
# Poisson regression using glm (Frequentist)
poisson_glm <- glm(y ~ x1 + x2, family = poisson(), data = df)
summary(poisson_glm)
Call:
glm(formula = y ~ x1 + x2, family = poisson(), data = df)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.94900 0.06650 14.271 < 2e-16 ***
x1 0.55868 0.06125 9.121 < 2e-16 ***
x2 -0.16575 0.05910 -2.804 0.00504 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 194.38 on 99 degrees of freedom
Residual deviance: 100.89 on 97 degrees of freedom
AIC: 370.19
Number of Fisher Scoring iterations: 5Poisson Regression (Bayesian)
# Poisson regression using brm (Bayesian)
poisson_brm <- brm(y ~ x1 + x2, family = poisson(), data = df)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.9e-05 seconds
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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Chain 4: Family: poisson
Links: mu = log
Formula: y ~ x1 + x2
Data: df (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 0.95 0.07 0.82 1.07 1.00 3002 2901
x1 0.56 0.06 0.43 0.68 1.00 2523 2455
x2 -0.17 0.06 -0.28 -0.05 1.00 3061 2407
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Beta Regression (Frequentist)
# Simulate some data for Beta regression
set.seed(540)
n <- 100
x <- rnorm(n)
y <- rbeta(n, shape1 = 2 + x, shape2 = 3 - x) # Beta-distributed
# Beta regression using betareg (Frequentist)
beta_glm <- betareg(y ~ x)
summary(beta_glm)
Call:
betareg(formula = y ~ x)
Quantile residuals:
Min 1Q Median 3Q Max
-2.5866 -0.6460 0.1194 0.7097 2.5844
Coefficients (mean model with logit link):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.56377 0.08954 -6.296 3.05e-10 ***
x 0.91834 0.10294 8.921 < 2e-16 ***
Phi coefficients (precision model with identity link):
Estimate Std. Error z value Pr(>|z|)
(phi) 4.759 0.640 7.436 1.03e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Type of estimator: ML (maximum likelihood)
Log-likelihood: 42.09 on 3 Df
Pseudo R-squared: 0.4843
Number of iterations: 13 (BFGS) + 1 (Fisher scoring) Beta Regression (Bayesian)
# Beta regression using brm (Bayesian)
beta_brm <- brm(y ~ x, family = Beta(), data = data.frame(x, y))Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 4.5e-05 seconds
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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Chain 2: Gradient evaluation took 1e-05 seconds
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
Chain 3:
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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Chain 4: Gradient evaluation took 9e-06 seconds
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Chain 4: Family: beta
Links: mu = logit; phi = identity
Formula: y ~ x
Data: data.frame(x, y) (Number of observations: 99)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.56 0.09 -0.74 -0.38 1.00 2776 2494
x 0.92 0.10 0.72 1.12 1.00 3066 3002
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 4.70 0.63 3.54 6.03 1.00 3097 2683
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Robust Student-t Regression (Bayesian)
# Simulate some data for Student-t regression
set.seed(123)
n <- 100
x1 <- rnorm(n)
y <- 5 + 2 * x1 + rt(n, df = 3) # Simulate t-distributed errors
df <- data.frame(y,x1)
# Student-t regression using brm (Bayesian)
student_brm <- brm(y ~ x1, family = student(), data = df)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.4e-05 seconds
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
Chain 3:
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SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 1.1e-05 seconds
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Chain 4: Family: student
Links: mu = identity; sigma = identity; nu = identity
Formula: y ~ x1
Data: df (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 5.01 0.12 4.77 5.25 1.00 3898 2659
x1 1.79 0.13 1.54 2.04 1.00 3843 2743
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.91 0.11 0.70 1.14 1.00 3024 2595
nu 2.05 0.48 1.30 3.14 1.00 2978 2134
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).